¹

1 Introduction↩︎

First, let us recall the classical \(L^p\)-Hardy inequality on the usual Euclidean space \(\mathbb{R}^n\): let \(1<p<n\) and \(f\in C_0^{\infty}(\mathbb{R}^n\backslash\{0\})\). Then, we have \[\begin{align} \label{class95multdim95hardy} \int_{\mathbb{R}^n}|\nabla f|^pdz\geq\left(\frac{n-p}{p}\right)^p\int_{\mathbb{R}^n}\frac{|f|^p}{|z|^p}dz, \end{align}\tag{1}\] where \(\left(\frac{n-p}{p}\right)^p\) is sharp but not achieved. The literature on the classical Hardy inequality (1 ) is enormous, and we are unable to list all the related articles in this paper. Therefore, we refer only to standard monographs such as [1]–[3]. Since in this paper we are mainly interested in the results regarding the Baouendi-Grushin operator, from this point onward, let us recall results only in this direction.

The first extension of the sharp Hardy inequality (1 , for \(p=2\), to the Baouendi-Grushin vector fields) was done by Garofalo [4]. To be more precise, the following inequality was obtained for all \(f \in C_0^{\infty}\left(\mathbb{R}^m \times \mathbb{R}^k \backslash\{(0,0)\}\right)\): \[\begin{align} \label{Gar} \int_{\mathbb{R}^n}|\nabla_{\gamma}f|^2 dz\geq \left(\frac{Q-2}{2}\right)^2 \int_{\mathbb{R}^n}\frac{|x|^{2 \gamma}}{\rho^{2\gamma+2}}|f|^2 dz \end{align}\tag{2}\] with \(n=m+k\), \(Q=m+(1+\gamma)k\) and \(\rho=\left(|x|^{2\gamma+2}+(1+\gamma)^2|y|^2\right)^{\frac{1}{2\gamma+2}}\). In this context, \(\nabla_{\gamma}=(\nabla_{x},|x|^{\gamma}\nabla_{y})\), where \(\nabla_{x}\) is the gradient operator in the variable \(x\), and \(\nabla_{y}\) is the gradient in the variable \(y\). The inequality (2 ) recovers the \(L^{2}\) version of (1 ) for \(\gamma=0\).

Following Garofalo’s influential work [4], there has been a considerable amount of research focused on developing inequalities of Hardy-type related to the Baouendi-Grushin operator, as demonstrated in publications such as [5]–[17].

For example, in D39A04?, D’Ambrosio was the first to obtain weighted \(L^p\)-versions of (2 ): let \(1<p<\infty\) and \(\Omega\) be an open subset of \(\mathbb{R}^n\). Then, we have \[\begin{align} \label{D39A04} \int_{\Omega}|x|^{\beta-\gamma p} \rho^{(1+\gamma) p-\alpha} \left|\nabla_\gamma f\right|^p dz \geq c^{p}_{Q,p,\alpha,\beta} \int_{\Omega} \frac{|x|^\beta}{\rho^\alpha} |f|^p dz \end{align}\tag{3}\] for all \(f\in D^{1,p}_{\gamma}(\Omega,|x|^{\beta-\gamma p}\rho^{(1+\gamma)p-\alpha})\), \(p>1\), \(k,m\geq1\), \(\alpha,\beta\in\mathbb{R}\) such that \(m+(1+\gamma)k>\alpha-\beta-p\) and \(m>\gamma p-\beta\). The space \(D^{1,p}_{\gamma}(\Omega, \omega)\) denotes the closure of \(C^{\infty}_{0}(\Omega)\) in the norm \((\int_{\Omega}^{}|\nabla_{\gamma}f|^{p}\omega dz)^{\frac{1}{p}}\) for \(\omega\in L^{1}_{loc}(\Omega)\) with \(\omega>0\) a.e. on \(\Omega\). The constant \(c^{p}_{Q,p,\alpha,\beta}\) in (3 ) is equal to \(\left(\frac{Q+\beta-\alpha}{p}\right)^p\) and sharp when \(0\in\Omega\).

Around the same time, Niu, Chen and Han [6] utilized the Picone identity to prove a variety of Hardy-type inequalities for the whole space, the open pseudo-ball and the external domain of the pseudo-ball.

Later, D’Ambrosio D39A05? showed Hardy-type inequalities related to quasilinear second-order degenerate elliptic differential operators. Specifically, the following inequality was obtained: let \(\alpha<-1\), \(p=Q>1\) and \(R>0\). Then, we have \[\begin{align} \label{D39A05} \int_{B_{R}}\left(\log\frac{R}{\rho}\right)^{\alpha+p}|\nabla_{\gamma}f|^{p}dz\geq\left(\frac{|\alpha+1|}{p}\right)^{p}\int_{B_{R}}\left(\log \frac{R}{\rho}\right)^{\alpha}\frac{|x|^{\gamma p}}{\rho^{\gamma p+p}}|f|^{p}dz \end{align}\tag{4}\] for all \(f\in D^{1,p}_{\gamma}\left(B_{R},\left(\log\frac{R}{\rho}\right)^{\alpha+p}\right)\) and \(B_{R}=\{z\in\mathbb{R}^n:\rho(z)<R\}\). Furthermore, the constant \(\left(\frac{|\alpha+1|}{p}\right)^p\), in (4 ), is sharp.

Shen and Jin, in [10], obtained Hardy-Rellich type inequalities with the optimal constant by a direct and simple approach. Then, in [11], Kombe proved weighted Hardy-type inequalities with remainder terms, which play a key role in establishing improved Rellich type inequalities. Yang, Su and Kong, in [12], derived improved Hardy inequalities on bounded domains containing the origin. In [14], Laptev, Ruzhansky and the first author of this paper obtained a refinement of the Hardy inequalities and derived weighted Hardy type inequalities for the quadratic form of the magnetic Baouendi-Grushin operator for the magnetic field of Aharonov-Bohm type. Also, Suragan and the first author of this paper, in [15], established a sharp remainder formula for the Poincaré inequality with a straightforward proof without using the variational principle. Recently, D’Arca in [16] derived Poincaré inequality and Hardy improvements related to some degenerate elliptic differential operators, including the Baouendi-Grushin operator. Then, Ganguly, Jotsaroop and Roychowdhury [17] showed Hardy, Hardy-Rellich, and Rellich identities and inequalities with sharp constants via spherical vector fields for \(\gamma=1\) and \(k=1\) in the setting of \(L^{2}\). On top of that, such inequalities have also been investigated for other sub-elliptic operators of various types (see, e.g. [4], [5], [7], [8], [16], [18], [19]).

Nevertheless, the result that we are most interested in is the one obtained by Kombe and Yener [13]. There they derived a sufficient condition on a pair of non-negative weight functions \(v\) and \(w\) in \(\mathbb{R}^{n}\) so that the general weighted Hardy type inequality with a remainder term is satisfied: let \(v \in C^1\left(\mathbb{R}^n\right)\) and \(w \in L_{\text{loc }}^1\left(\mathbb{R}^n\right)\) be nonnegative functions and \(\phi \in\) \(C^{\infty}\left(\mathbb{R}^n\right)\) be a positive function satisfying the differential inequality \[\begin{align} -\nabla_\gamma \cdot\left(v\left|\nabla_\gamma \phi\right|^{p-2} \nabla_\gamma\phi\right) \geq w\phi^{p-1} \end{align}\] a.e. in \(\mathbb{R}^n\). There is a positive number \(c_p=c(p)\) such that; if \(p \geq 2\), then \[\begin{align} \label{kom1} \int_{\mathbb{R}^n} v\left|\nabla_\gamma f\right|^p dz \geq \int_{\mathbb{R}^n} w|f|^p dz+c_p \int_{\mathbb{R}^n} v\left|\nabla_\gamma \frac{f}{\phi}\right|^p \phi^p dz \end{align}\tag{5}\] and if \(1<p<2\), then \[\begin{align} \label{kom2} \int_{\mathbb{R}^n} v\left|\nabla_\gamma f\right|^p dz \geq \int_{\mathbb{R}^n} w|f|^p dz+c_p \int_{\mathbb{R}^n} \frac{v\left|\nabla_\gamma \frac{f}{\phi}\right|^2 \phi^2}{\left(\left|\frac{f}{\phi} \nabla_\gamma \phi\right|+\left|\nabla_\gamma \frac{f}{\phi}\right| \phi\right)^{2-p}} dz \end{align}\tag{6}\] for all real-valued functions \(f \in C_0^{\infty}\left(\mathbb{R}^n\right)\).

The main purpose of this paper is to provide similar results, but for complex-valued functions and nonnegative explicit remainder terms. In particular, we obtain the following refined general weighted \(L^{p}\)-Hardy type identity: let \(1<p<\infty\) and let \(\Omega\subseteq\mathbb{R}^{m+k}\) be an open set such that the integrals below make sense. Let \(v\in C^{1}(\Omega\backslash\Sigma)\) and \(w\in L^{1}_{loc}(\Omega\backslash\Sigma)\) be nonnegative functions satisfying the condition (in the weak sense) \[\begin{align} \label{main32condt32intro} 0\leq\phi:=\nabla_{\gamma}\cdot \left(w^{\frac{p-1}{p}}v^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right)-pw \end{align}\tag{7}\] a.e. in \(\Omega\backslash\Sigma\). Then, for all complex-valued \(f\in C^{\infty}_0(\Omega\backslash\Sigma)\), we have \[\begin{align} \label{main32res32intro} \int_{\Omega}^{}v\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f|^{p}}{|\nabla_{\gamma}\rho|^{p}}dz=\int_{\Omega}^{}w|f|^{p}dz+\int_{\Omega}^{}C_{p}(\xi,\eta)dz+\int_{\Omega}\phi|f|^{p}dz, \end{align}\tag{8}\] where the functional \(C_p(\cdot,\cdot)\) is given by \[\begin{align} \label{cp32formula32intro} C_p(\xi,\eta):=|\xi|^p-|\xi-\eta|^p-p|\xi-\eta|^{p-2}\textrm{Re}(\xi-\eta)\cdot\overline{\eta}\geq0 \end{align}\tag{9}\] and \[\begin{align} \label{main32notation32intro} \xi:=v^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|},\quad \eta:=v^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|} + w^{\frac{1}{p}}f \end{align}\tag{10}\] with \(\Sigma\) being the set of singular points of \(v\) and \(w\).

The result (8 ) improves the left-hand side of (5 ) and (6 ) by replacing \(\nabla_{\gamma}f\) with \(\frac{\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f}{|\nabla_{\gamma}\rho|}\). Here, we also have nonnegative explicit remainder terms. Moreover, (8 ) holds true for all complex-valued functions, whereas (5 ) and (6 ) are valid only for real-valued functions.

A wide range of weighted Hardy-type identities can be derived by selecting appropriate weights \(w\) and \(v\) that satisfy the required condition (for more details, see Section 4.1). For example, in the special case, when \(v=|x|^{\beta-\gamma p}\rho^{p(1+\gamma)-\alpha}\) and \(w=\left(\frac{Q+\beta-\alpha}{p}\right)^{p}\frac{|x|^{\beta}}{\rho^{\alpha}}\), in (7 ), we have \(\phi=0\), and a sharp remainder formula of (3 ) is obtained with \(\frac{\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f}{|\nabla_{\gamma}\rho|}\): let \(1<p<\infty\), \(m,k\geq1\) and \(\alpha,\beta\in \mathbb{R}\) be such that \(Q> \alpha-\beta\). Then, we have \[\begin{gather} \label{dam32ref} \int_{\mathbb{R}^n}|x|^{\frac{\beta-\gamma p}{p}}\rho^{\frac{p(1+\gamma)-\alpha}{p}}\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f|^p}{|\nabla_{\gamma}\rho|^p}dz=\left(\frac{Q+\beta-\alpha}{p}\right)^{p}\int_{\mathbb{R}^n}\frac{|x|^{\beta}}{\rho^{\alpha}}|f|^pdz \\+\int_{\mathbb{R}^n}C_{p}\biggl(|x|^{\frac{\beta-\gamma p}{p}}\rho^{\frac{p(1+\gamma)-\alpha}{p}}\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|},|x|^{\frac{\beta-\gamma p}{p}}\rho^{\frac{p(1+\gamma)-\alpha}{p}}\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|} \\+ \left(\frac{Q+\beta-\alpha}{p}\right)\frac{|x|^{\frac{\beta}{p}}}{\rho^{\frac{\alpha}{p}}}f\biggr)dz \end{gather}\tag{11}\] for all complex-valued \(f\in C^{\infty}_0(\mathbb{R}^m\times\mathbb{R}^{k}\backslash\{(0,0)\})\). Here, we can use the results of Cazacu, Krejčiřı́k, Lam and Laptev [20] to obtain the estimates of the remainder term from below for \(p\geq2\) (see Corollary 13). Additionally, due to the recent results of Chen and Tang [21] we can derive (11 ) with different sharp remainder terms for \(1<p<2\leq n\) (see Corollary 14). Moreover, when \(\beta=\gamma p\) and \(\alpha=p(1+\gamma)\) in (11 ), the following identity is derived: let \(1<p<\infty\) and \(\gamma \geq0\). Then, for any complex-valued \(f\in C^{\infty}_{0}(\mathbb{R}^m\times\mathbb{R}^{n}\backslash\{(0,0)\})\), there holds \[\begin{gather} \label{gar32ref32intro32lp} \int_{\mathbb{R}^n}\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f|^{p}}{|\nabla_{\gamma}\rho|^{p}}dz= \left(\frac{Q-p}{p}\right)^{p}\int_{\mathbb{R}^n}\frac{|x|^{\gamma p}}{\rho^{\gamma p + p}}|f|^{p}dz \\+\int_{\mathbb{R}^n}C_{p}\left(\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|},\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|}+\left(\frac{Q-p}{p}\right)\frac{|x|^{\gamma}}{\rho^{\gamma+1}}f\right)dz. \end{gather}\tag{12}\] By setting \(\gamma=0\) in (12 ), we obtain a refinement of the classical Hardy inequality (1 ), which was also obtained, as a special case, by Kalaman and the first author of this article in [22] under certain parameters: let \(1<p<\infty\). Then, for any complex-valued functions \(f\in C^{\infty}_{0}(\mathbb{R}^n\backslash\{0\})\), we have \[\begin{gather} \int_{\mathbb{R}^n}\frac{|z\cdot\nabla f|^{p}}{|z|^{p}}dz=\left(\frac{n-p}{p}\right)^{p}\int_{\mathbb{R}^n}\frac{|f|^{p}}{|z|^{p}}dz\\+\int_{\mathbb{R}^n}C_{p}\left(\frac{z\cdot\nabla f}{|z|},\frac{z\cdot\nabla f}{|z|}+\left(\frac{n-p}{p}\right)f\right)dz. \end{gather}\] We refer to [23] for the case of real-valued functions.

In addition, the identity (12 ) allows us to obtain a refinement of (2 ) for \(p=2\): let \(\gamma\geq0\). Then, for any complex-valued \(f\in C^{\infty}_{0}(\mathbb{R}^m\times\mathbb{R}^{n}\backslash\{(0,0)\})\), we have \[\begin{gather} \int_{\mathbb{R}^n}\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f|^{2}}{|\nabla_{\gamma}\rho|^{2}}dz= \left(\frac{Q-2}{2}\right)^{2}\int_{\mathbb{R}^n}\frac{|x|^{2\gamma}}{\rho^{2\gamma+2}}|f|^{2}dz \\+\int_{\mathbb{R}^n}C_{2}\left(\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|},\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|}+\left(\frac{Q-2}{2}\right)\frac{|x|^{\gamma}}{\rho^{\gamma+1}}f\right)dz. \end{gather}\]

Another interesting application of our main result (8 ) is the refinement of the power logarithmic \(L^p\)-Hardy type inequality (4 ). That is, setting \(v=\left(\log \frac{R}{\rho}\right)^{\alpha+p}\) and \(w=\left(\frac{|\alpha+1|}{p}\right)^{p}\left(\log \frac{R}{\rho}\right)^{\alpha}\frac{|x|^{\gamma p}}{\rho^{\gamma p+p}}\), in (7 ), gives the following sharp remainder formula of (4 ) with \(\frac{\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f}{|\nabla_{\gamma}\rho|}\): let \(1<p<\infty\), \(\alpha\in\mathbb{R}\) and \(R>0\) be such that \(\alpha+1<0\). Then, for all complex-valued \(f\in C^{\infty}_{0}(B_{R}\backslash\{(0,0)\})\), we have \[\begin{gather} \label{cor432eq32intro} \int_{B_{R}}\left(\log\frac{R}{\rho}\right)^{\alpha+p}\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f|^p}{|\nabla_{\gamma}\rho|^p}dz=\left(\frac{|\alpha+1|}{p}\right)^{p}\int_{B_{R}}\left(\log \frac{R}{\rho}\right)^{\alpha}\frac{|x|^{\gamma p}}{\rho^{\gamma p+p}}|f|^{p}dz\\+\int_{B_{R}}C_{p}\Biggl(\left(\log\frac{R}{\rho}\right)^{\frac{\alpha+p}{p}}\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|},\left(\log\frac{R}{\rho}\right)^{\frac{\alpha+p}{p}}\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|} \\+\left(\frac{|\alpha+1|}{p}\right)\left(\log\frac{R}{\rho}\right)^{\frac{\alpha}{p}}\frac{|x|^{\gamma }}{\rho^{\gamma+1}}f\Biggr)dz \\+\left(\frac{|\alpha+1|}{p}\right)^{p-1}(Q-p)\int_{B_{R}}\frac{|x|^{\gamma p}}{\rho^{\gamma p + p}}\left(\log \frac{R}{\rho}\right)^{\alpha+1}|f|^{p}dz. \end{gather}\tag{13}\] If we set \(\gamma=0\), \(\alpha=-n\) and \(p=n\), in (13 ), then for \(n\geq2\), we obtain the following scale-invariant critical version of the Hardy inequality derived by Ioku, Ishiwata and Ozawa [24], [25]: \[\begin{gather} \int_{B_{R}}\frac{|z\cdot\nabla f|^{n}}{|z|^{n}}dz=\left(\frac{n-1}{n}\right)^{n}\int_{B_R}\frac{|f|^{n}}{|z|^{n}\left(\log \frac{R}{|z|}\right)^n}dz\\+\int_{B_{R}}C_{n}\left(\frac{z\cdot\nabla f}{|z|},\frac{z\cdot\nabla f}{|z|}+\left(\frac{n-1}{n}\right)\frac{f}{|z|\left(\log \frac{R}{|z|}\right)}\right)dz \end{gather}\] for all complex-valued \(f\in C^{\infty}_{0}(B_{R}\backslash\{(0,0)\})\).

We also apply the main result to obtain the general weighted Caffarelli-Kohn-Nirenberg (CKN) type inequalities with remainder terms, which, in special cases, give Heisenberg-Pauli-Weyl (HPW) type inequalities. Before we state the results, let us recall the classical CKN inequality from [26]: let \(n \in \mathbb{N}\) and let \(p,\;q,\;r,\;a,\;b,\;d,\;\delta \in \mathbb{R}\) such that \(p, q \geq 1,\;r > 0,\;0 \leq \delta \leq 1\), and \[\begin{align} \frac{1}{p} + \frac{a}{n}, \quad \frac{1}{q} + \frac{b}{n}, \quad \frac{1}{r} + \frac{c}{n} > 0, \end{align}\] where \(c = \delta d + (1 - \delta)b\). Then there exists a positive constant \(C\) such that \[\begin{align} \label{ckn} \left\| |x|^a |\nabla f| \right\|^{\delta}_{L^p(\mathbb{R}^n)} \left\| |x|^b f \right\|_{L^q(\mathbb{R}^n)}^{1 - \delta}\geq C\left\| |x|^c f \right\|_{L^r(\mathbb{R}^n)}, \end{align}\tag{14}\] holds for all \(f \in C_0^\infty(\mathbb{R}^n)\) if and only if the following conditions hold: \[\begin{align} \frac{1}{r} + \frac{c}{n} = \delta \left( \frac{1}{p} + \frac{a - 1}{n} \right) + (1 - \delta)\left( \frac{1}{q} + \frac{b}{n} \right), \end{align}\] \[\begin{align} a - d \geq 0 \quad \text{if} \quad \delta > 0, \end{align}\] \[\begin{align} a - d \leq 1 \quad \text{if} \quad \delta > 0 \quad \text{and} \quad \frac{1}{r} + \frac{c}{n} = \frac{1}{p} + \frac{a - 1}{n}. \end{align}\] The classical CKN inequalities (14 ) have been extended to a variety of settings, including the Baouendi-Grushin vector fields, general homogeneous Carnot groups and the Heisenberg group [13], [17], [27], [28]. On the anisotropic analogues of the classical \(L^{p}\)-CKN inequalities, we refer to [29], [30]. We also refer to other extended CKN inequalities as well as various versions and applications [31]–[38].

Under specific choice of parameters, the classical CKN inequalities (14 ) imply the classical HPW uncertainty principle [39]: \[\begin{align} \label{hpw} \left( \int_{\mathbb{R}^n} |\nabla f|^2 \, dz \right) \left( \int_{\mathbb{R}^n} |z|^2 |f|^2 \, dz \right) \geq C \left( \int_{\mathbb{R}^n} |f|^2 \, dz \right)^2 \end{align}\tag{15}\] for all \(f \in C_0^\infty(\mathbb{R}^n)\). The inequality (15 ) mathematically expresses the fundamental physical principle that, in any quantum state, it is impossible to simultaneously determine both a particle’s position and momentum with complete accuracy.

Now we are ready to state our results regarding general weighted CKN type inequalities with remainder terms: let \(1<p,q<\infty\), \(0<r<\infty\) with \(p+q\geq r\), \(\delta \in [0,1] \cap \left[ \frac{r - q}{r}, \frac{p}{r} \right]\) and \(b, c \in \mathbb{R}\). Assume that \(\frac{\delta r}{p} + \frac{(1-\delta) r}{q} = 1\) and \(c = \frac{\delta}{p} + b(1 - \delta)\). Then, for any complex-valued \(f\in C^{\infty}_{0}(\mathbb{R}^m\times\mathbb{R}^{n}\backslash\{(0,0)\})\), we have \[\begin{gather} \Biggl[\left\lVert v^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f}{|\nabla_{\gamma}\rho|}\right\rVert^{p}_{L^{p}(\mathbb{R}^n)}\\-\int_{\mathbb{R}^n}C_{p}\left(v^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|},v^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|} + w^{\frac{1}{p}}f\right)dz\Biggr]^{\frac{\delta}{p}}\left\lVert w^{b}f\right\rVert^{1-\delta}_{L^{q}(\mathbb{R}^n)}\geq\left\lVert w^{c}f\right\rVert_{L^{r}(\mathbb{R}^n)}. \end{gather}\]

After choosing appropriate weights and parameters, we obtain the following HPW type inequalities: let \(\gamma\geq0\). Then, we have \[\begin{align} \label{almost32hpw} \left(\int_{\mathbb{R}^n}\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f|^{2}}{|\nabla_{\gamma}\rho|^{2}}dz\right)\left(\int_{\mathbb{R}^n}\frac{\rho^{2\gamma+2}}{|x|^{2\gamma}}|f|^{2}dz\right)\geq\left(\frac{Q-2}{2}\right)^{2}\left(\int_{\mathbb{R}^n}|f|^{2}dz\right)^{2} \end{align}\tag{16}\] for any complex-valued \(f\in C^{\infty}_0(\mathbb{R}^m\times\mathbb{R}^{k}\backslash\{(0,0)\})\).

Substituting \(\gamma=0\) and applying the Cauchy-Schwarz inequality, in (16 ), we obtain the uncertainty principle (15 ) with an explicit constant: \[\begin{align} \left( \int_{\mathbb{R}^n} |\nabla f|^2 \, dz \right) \left( \int_{\mathbb{R}^n} |z|^2 |f|^2 \, dz \right) \geq \frac{(n-2)^2}{4} \left( \int_{\mathbb{R}^n} |f|^2 \, dz \right)^2. \end{align}\]

In Section 2, we briefly review the essential definitions, notations and preliminary results. Section 3 is devoted to establishing the main refined general weighted \(L^p\)‑Hardy type inequalities and identities for the range \(1 < p < \infty\). Furthermore, we provide proofs of remainder estimates for \(p\geq2\) and \(1<p<2\leq n\) under certain conditions. Finally, in Section 4, we explore applications of these results. Section 4.1 presents sharp refinements of various classical and weighted Hardy inequalities on \(\mathbb{R}^n\), within \(\rho\)‑ball and involving logarithmic weights. In Section 4.2, we derive general weighted CKN type inequalities with remainder terms, which in turn yield HPW type uncertainty principles.

2 Preliminaries↩︎

In this section, we will provide the notation and some preliminary results. Suppose \(z=(x_{1},\ldots,x_{m},y_{1},\ldots,y_{k})\) or simply \(z=(x,y)\) \(\in\mathbb{R}^{m}\times\mathbb{R}^{k}\) with \(m+k=n\) and \(m,k\geq1\). The sub-elliptic gradient is the \(n\)-dimensional vector field given by \[\nabla_{\gamma}=(X_{1},\ldots,X_{m},Y_{1},\ldots,Y_{k}).\] Here, the components are defined as: \[X_{i}=\frac{\partial}{\partial{x_i}}, \quad i=1,\ldots,m, \quad Y_{j}=|x|^{\gamma}\frac{\partial}{\partial y_{j}}, \quad j=1,\ldots,k.\] The Baouendi-Grushin operator on \(\mathbb{R}^{n}\) is the operator \[\Delta_{\gamma} = \nabla_{\gamma}\cdot\nabla_{\gamma}=\Delta_{x}+|x|^{2\gamma}\Delta_{y},\] where \(\Delta_{x}\) and \(\Delta_{y}\) are Laplace operators in the variables \(x\in\mathbb{R}^{m}\) and \(y\in\mathbb{R}^{k}\), respectively. The Baouendi-Grushin operator is a sum of squares of \(C^{\infty}\) vector fields satisfying the Hörmander condition for even positive integers \(\gamma\): \[\begin{align} \operatorname{rank} \operatorname{Lie}\left[X_1, \ldots, X_m, Y_1, \ldots, Y_k\right]=n \text{. } \nonumber \end{align}\] The anisotropic dilation attached to \(\Delta_{\gamma}\) is defined as follows: \[\begin{align} \delta_\lambda(x, y)=\left(\lambda x, \lambda^{1+\gamma} y\right), \quad \lambda>0, \quad z=(x,y)\in\mathbb{R}^{m+k}. \nonumber \end{align}\] The dilation’s homogeneous dimension is given by \[\begin{align} Q=m+(1+\gamma) k. \nonumber \end{align}\] The Lebesgue measure’s formula for alternating variables suggests that \[\begin{align} d \circ \delta_\lambda(x, y)=\lambda^Q d x d y. \nonumber \end{align}\] It is straightforward to check that \[\begin{align} X_i\left(\delta_\lambda\right)=\lambda \delta_\lambda\left(X_i\right), \quad Y_i\left(\delta_\lambda\right)=\lambda \delta_\lambda\left(Y_i\right) \nonumber \end{align}\] and hence \[\begin{align} \nabla_\gamma \circ \delta_\lambda=\lambda \delta_\lambda \nabla_\gamma. \nonumber \end{align}\] The respective distance function from the origin to some point in space \(z=(x,y)\in\mathbb{R}^{n}\): \[\begin{align} \rho=\rho(z):=\left(|x|^{2(1+\gamma)}+(1+\gamma)^2|y|^2\right)^{\frac{1}{2(1+\gamma)}} . \nonumber \end{align}\] The function \(\rho\) is positive, smooth away from the origin and symmetric. One can verify that \[\begin{align} \nabla_{\gamma}\rho=\left(\frac{|x|^{2\gamma}x}{\rho^{2\gamma+1}},\frac{(1+\gamma)|x|^{\gamma}y}{\rho^{2\gamma+1}}\right), \end{align}\] which gives us \[\begin{align} |\nabla_{\gamma}\rho|=\frac{|x|^\gamma}{\rho^\gamma}. \end{align}\] By direct calculation, we also get \[\begin{align} \label{main32formula} \nabla_{\gamma}\cdot\left(\rho^{c}|x|^{s}\nabla_{\gamma}\rho\right)=(Q+c+s-1)\frac{|x|^{2\gamma+s}}{\rho^{2\gamma+1-c}} \end{align}\tag{17}\] with \(c,s\in\mathbb{R}\), which by setting \(c=\gamma\) and \(s=-\gamma\) gives \[\begin{align} \label{formula32for32term2} \nabla_{\gamma}\left(\frac{\rho^{\gamma}}{|x|^{\gamma}}\nabla_{\gamma}\rho\right)=\nabla_{\gamma}\left(\frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right)=(Q-1)\frac{|x|^{\gamma}}{\rho^{\gamma+1}}. \end{align}\tag{18}\] We define \(B_{R}=\{z\in\mathbb{R}^n:\rho(z)<R\}\) as a \(\rho\)-ball.

3 Refined general weighted \(L^p\)-Hardy type inequalities and identities↩︎

In this section, we prove the improved general weighted \(L^p\)-Hardy type inequalities and identities related to the Baouendi-Grushin operator.

\[\begin{align} \label{cancel32term} \int_{\Omega}^{}w|f|^{p}dz&= \frac{1}{p}\int_{\Omega}^{}|f|^{p}\text{div}_{\nabla_{\gamma}}\left(w^{\frac{p-1}{p}}v^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right)dz-\frac{1}{p}\int_{\Omega}\phi |f|^{p}dz \nonumber \\&=-\text{Re}\int_{\Omega}^{}f|f|^{p-2}\frac{\overline{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}}{|\nabla_{\gamma}\rho|}w^{\frac{p-1}{p}}v^{\frac{1}{p}}dz-\frac{1}{p}\int_{\Omega}\phi|f|^{p}dz. \end{align}\tag{19}\] By Hölder inequality, we get \[\begin{align} \int_{\Omega}w|f|^{p}dz&\leq\int_{\Omega}|f|^{p-1}w^{\frac{p-1}{p}}v^{\frac{1}{p}}\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f|}{|\nabla_{\gamma}\rho|}dz \nonumber \\&\leq\left(\int_{\Omega}w|f|^{p}dz\right)^{\frac{p-1}{p}}\left(\int_{\Omega}v\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f|^{p}}{|\nabla_{\gamma}\rho|^{p}}dz\right)^{\frac{1}{p}}. \end{align}\] This immediately gives us Part (1) of Theorem 1. Now for Part (2), we utilize the notation (10 ) and formula (9 ): \[\begin{align} &C_{p}(\xi,\eta)=v\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f|^{p}}{|\nabla_{\gamma}\rho|^{p}}-\left|-w^{\frac{1}{p}}f\right|^{p}-p\left|-w^{\frac{1}{p}}f\right|^{p-2}\text{Re}\left(-w^{\frac{1}{p}}f\right) \\&\times\overline{\left(v^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|} + w^{\frac{1}{p}}f\right)}. \end{align}\] Expanding and simplifying results in \[\begin{align} &C_{p}(\xi,\eta)=v\frac{|\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f|^{p}}{|\nabla_{\gamma}\rho|^{p}}-w|f|^{p}+p\text{Re}w^{\frac{p-1}{p}}f|f|^{p-2}\overline{v^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|}}+pw|f|^{p}. \end{align}\] Integrating both sides, we have \[\begin{align} &\int_{\Omega}^{}C_{p}(\xi,\eta)dz=\int_{\Omega}^{}v\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f|^{p}}{|\nabla_{\gamma}\rho|^{p}}dz-\int_{\Omega}^{}w|f|^{p}dz \\&+p\text{Re}\int_{\Omega}^{}f|f|^{p-2}\frac{\overline{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}}{|\nabla_{\gamma}\rho|}w^{\frac{p-1}{p}}v^{\frac{1}{p}}dz+p\int_{\Omega}^{}w|f|^{p}dz. \end{align}\] Using the identity (19 ): \[\begin{align} &\int_{\Omega}^{}C_{p}(\xi,\eta)dz=\int_{\Omega}^{}v\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f|^{p}}{|\nabla_{\gamma}\rho|^{p}}dz-\int_{\Omega}^{}w|f|^{p}dz \\&+p\text{Re}\int_{\Omega}^{}f|f|^{p-2}\frac{\overline{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}}{|\nabla_{\gamma}\rho|}w^{\frac{p-1}{p}}v^{\frac{1}{p}}dz+p\int_{\Omega}^{}w|f|^{p}dz \\&=\int_{\Omega}^{}v\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f|^{p}}{|\nabla_{\gamma}\rho|^{p}}dz-\int_{\Omega}^{}w|f|^{p}dz \\&-p\int_{\Omega}^{}w|f|^{p}dz-\int_{\Omega}\phi|f|^{p}dz+p\int_{\Omega}^{}w|f|^{p}dz \\&=\int_{\Omega}^{}v\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f|^{p}}{|\nabla_{\gamma}\rho|^{p}}dz-\int_{\Omega}^{}w|f|^{p}dz-\int_{\Omega}\phi|f|^{p}dz. \end{align}\] As a result, we obtain (?? ). \(\square\)

If the condition (?? ) is satisfied with \(\phi=0\) in (?? ), then we obtain the remainder estimate from below for \(p\geq2\) via the result from [20]:

Additionally, under the same condition \(\phi=0\) in (?? ), recent results from [21] allow us to obtain (?? ) with different remainder terms for \(1<p<2\leq n\):

Before proving Theorem 2 and 3, we first present the following lemmata from [20] and [21], which play an important role in the argument:

\[\begin{align} \int_{\Omega}^{}C_{p}(\xi,\eta)dz=\int_{\Omega}^{}v\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f|^{p}}{|\nabla_{\gamma}\rho|^{p}}dz-\int_{\Omega}^{}w|f|^{p}dz. \end{align}\] Using Lemma 4, we derive (?? ). \(\square\)

4 Applications of Theorem 1, 2 and 3↩︎

Let \(\epsilon>0\) be given. Define \[\begin{align} \rho_{\epsilon}:=\left(|x|^{2(1+\gamma)}_{\epsilon}+(1+\gamma)^{2}|y|^{2}\right)^{\frac{1}{2(1+\gamma)}}, \end{align}\] where \(|x|_{\epsilon}:=\left(\epsilon^{2}+\sum_{i=1}^{m}x^{2}_{i}\right)^{\frac{1}{2}}\). To ensure the rigor of the following arguments, we need to replace \(\rho\) with its regularization \(\rho_{\epsilon}\) and take the limit \(\epsilon\longrightarrow 0\) after performing the computations. However, for simplicity, we proceed with a formal approach.

4.1 Refinements of weighted \(L^{p}\)-Hardy type inequalities and identities.↩︎

\[\begin{align} v = 1, \quad w=\left(\frac{p-1}{p}\right)^p\frac{|x|^{\gamma p}}{(R-\rho)^{p}\rho^{\gamma p}} \end{align}\] then, checking the condition (?? ), we see that it actually holds and, thus, we have the following refined inequalities and identities of the result by Niu, Chen and Han [6]:

\[\begin{align} &\nabla_{\gamma} \cdot\left(\left(\frac{p-1}{p}\right)^{p\frac{p-1}{p}}\left(\frac{|x|^{\gamma p}}{(R-\rho)^{p}\rho^{\gamma p}}\right)^{\frac{p-1}{p}}\frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right)=\nabla_{\gamma}\cdot\left(\Phi \frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right) \nonumber \\&=\nabla_{\gamma}\Phi \cdot \frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}+\Phi \nabla_{\gamma}\cdot \left(\frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right) \nonumber \\&=\frac{1}{|\nabla_{\gamma}\rho|}\nabla_{\gamma}\Phi \cdot \nabla_{\gamma}\rho + \Phi\nabla_{\gamma}\cdot \left(\frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right) \nonumber \\&=T_1+T_2,\label{cor132t143t2} \end{align}\tag{20}\] where \[\begin{align} &T_1=\frac{1}{|\nabla_{\gamma}\rho|}\nabla_{\gamma}\Phi \cdot \nabla_{\gamma}\rho, \nonumber \\&T_2=\Phi\nabla_{\gamma}\cdot \left(\frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right) \label{cor132t2} \end{align}\tag{21}\] with \[\begin{align} &\Phi=\left[\left(\frac{p-1}{p}\right)^{p}\frac{|x|^{\gamma p}}{(R-\rho)^{p}\rho^{\gamma p}}\right]^{\frac{p-1}{p}} \\&=\left(\frac{p-1}{p}\right)^{p-1}\frac{|x|^{\gamma(p-1)}}{(R-\rho)^{p-1}\rho^{\gamma(p-1)}}. \end{align}\] Substituting formula (18 ) to (21 ), we get \[\begin{align} &T_2=\Phi \cdot\left[(Q-1)\frac{|x|^{\gamma}}{\rho^{\gamma+1}}\right] \nonumber \\&=\left(\frac{p-1}{p}\right)^{p-1}\frac{|x|^{\gamma(p-1)}}{(R-\rho)^{p-1}\rho^{\gamma(p-1)}}\cdot(Q-1)\frac{|x|^{\gamma}}{\rho^{\gamma+1}} \nonumber \\&=\left(\frac{p-1}{p}\right)^{p-1}(Q-1)\frac{|x|^{\gamma p}}{(R-\rho)^{p-1}\rho^{\gamma p+1}}. \label{cor132t2951} \end{align}\tag{22}\] Now we compute \(T_1\): \[\begin{align} &T_1=\frac{1}{|\nabla_{\gamma}\rho|}(\nabla_{\gamma}\Phi)\cdot \nabla_{\gamma}\rho \nonumber \\&=\frac{1}{|\nabla_{\gamma}\rho|}\left(\frac{p-1}{p}\right)^{p-1}\nabla_{\gamma}\left((R-\rho)^{-(p-1)}\left(\frac{|x|^{\gamma}}{\rho^{\gamma}}\right)^{p-1}\right)\cdot\nabla_{\gamma}\rho \nonumber \\&=\frac{\rho^{\gamma}}{|x|^{\gamma}}\left(\frac{p-1}{p}\right)^{p-1}(p-1)(R-\rho)^{-p}\frac{|x|^{\gamma(p+1)}}{\rho^{\gamma(p+1)}} \nonumber \\&=p\left(\frac{p-1}{p}\right)^{p}\frac{|x|^{\gamma p}}{(R-\rho)^{p}\rho^{\gamma p}}. \label{cor132t1951} \end{align}\tag{23}\] Putting (22 ) and (23 ) to (20 ), we obtain \[\begin{align} &T_{1}+T_{2}=p\left(\frac{p-1}{p}\right)^{p}\frac{|x|^{\gamma p}}{(R-\rho)^{p}\rho^{\gamma p}}+\left(\frac{p-1}{p}\right)^{p-1}(Q-1)\frac{|x|^{\gamma p}}{(R-\rho)^{p-1}\rho^{\gamma p+1}} \\&=pw+\left(\frac{p-1}{p}\right)^{p-1}(Q-1)\frac{|x|^{\gamma p}}{(R-\rho)^{p-1}\rho^{\gamma p+1}}. \end{align}\] This implies that \(\phi=\left(\frac{p-1}{p}\right)^{p-1}(Q-1)\frac{|x|^{\gamma p}}{(R-\rho)^{p-1}\rho^{\gamma p+1}}\geq0\). Therefore, the condition (?? ) is satisfied and the results (?? ) and (?? ) are obtained. Since the function \(h=\left(R-\rho\right)^{-\frac{p-1}{p}}\) satisfies the Hölder equality condition \[\begin{align} \left(\frac{p}{p-1}\right)^{p}\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}h|^{p}}{|\nabla_{\gamma}\rho|^{p}}=\frac{|x|^{\gamma p}}{(R-\rho)^{p}\rho^{\gamma p}}|h|^{p}, \end{align}\] the constant in the inequality (?? ) is sharp. \(\square\)

We are able to obtain the refinements of some results by D’Ambrosio in D39A04? and D39A05?. For instance, setting \[\begin{align} v=|x|^{\beta-\gamma p}\rho^{p(1+\gamma)-\alpha},\quad w=\left(\frac{Q+\beta-\alpha}{p}\right)^{p}\frac{|x|^{\beta}}{\rho^{\alpha}}, \end{align}\] we get the following refinements of the weighted \(L^p\)-Hardy type inequality D39A04?:

Substituting \(\beta=\gamma p\) and \(\alpha=p(1+\gamma)\) in (?? ), we obtain the following identity:

Since Corollary 9 gives \(\phi=0\) in (?? ), we can apply Theorem 2 to obtain remainder estimate of (?? ) from below for \(p\geq2\):

Furthermore, applying Theorem 3, we obtain (?? ) with different remainder terms for \(1<p<2\leq n\):

\[\begin{align} \label{damfor} &\nabla_{\gamma} \cdot\left(\left(\left(\frac{Q+\beta-\alpha}{p}\right)^{p}\frac{|x|^{\beta }}{\rho^{\alpha}}\right)^{\frac{p-1}{p}}|x|^{\frac{\beta-\gamma p}{p}}\rho^{\frac{p(1+\gamma)-\alpha}{p}}\frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right)\nonumber \\&=\nabla_{\gamma}\cdot\left(\left(\frac{Q+\beta-\alpha}{p}\right)^{p-1}|x|^{\beta-\gamma}\rho^{1+\gamma-\alpha}\frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right)\nonumber \\&=\left(\frac{Q+\beta-\alpha}{p}\right)^{p-1}\nabla_{\gamma}\cdot\left(|x|^{\beta-2\gamma}\rho^{1+2\gamma-\alpha}\nabla_{\gamma}\rho\right). \end{align}\tag{24}\] Using formula (17 ), in (24 ), with \(s=\beta-2\gamma\) and \(c=1+2\gamma-\alpha\), we get \[\begin{align} &\left(\frac{Q+\beta-\alpha}{p}\right)^{p-1}\nabla_{\gamma}\cdot\left(|x|^{\beta-2\gamma}\rho^{1+2\gamma-\alpha}\nabla_{\gamma}\rho\right)=\left(\frac{Q+\beta-\alpha}{p}\right)^{p-1}(Q+\beta-\alpha)\frac{|x|^{\beta}}{\rho^{\alpha}} \\&=p\left(\frac{Q+\beta-\alpha}{p}\right)^{p}\frac{|x|^{\beta}}{\rho^{\alpha}}=pw+0. \end{align}\] As we can see, the condition (?? ) is satisfied with \(\phi=0\), giving the desired results (?? ) and (?? ). Since the function \(h=\rho^{\left(\frac{Q+\beta-\alpha}{p}\right)}\) satisfies the Hölder equality condition, \[\begin{align} \left(\frac{p}{Q+\beta-\alpha}\right)^{p}|x|^{\beta-\gamma p}\rho^{(1+\gamma)p-\alpha}\frac{|\nabla_{\gamma}h\cdot\nabla_{\gamma}\rho|^{p}}{|\nabla_{\gamma}\rho|^{p}}=\frac{|x|^{\beta}}{\rho^{\alpha}}|h|^{p}, \end{align}\] the constant in the inequality (?? ) is sharp. \(\square\)

Setting

\[\begin{align} v = \frac{|\nabla_\gamma \rho|^{\alpha}}{\rho^{p(\theta - 1)}}, \quad w = \left(\frac{Q - p\theta}{p}\right)^{p}\frac{\left| \nabla_\gamma \rho \right|^{\alpha + p}}{\rho^{p\theta}} \end{align}\] gives us an extended version of D’Arca results [16] for \(1<p<\infty\), \(\alpha\in\mathbb{R}\) and for complex-valued \(f\in C^{\infty}_{0}(B_{R}\backslash\{(0,0)\})\):

Under \(\theta=1\) and \(\alpha=0\) in (?? ), we recover the result (?? ) on the \(\rho\)-ball \(B_{R}\):

In the same way as with (?? ), we have \(\phi=0\) in (?? ), which, by applying Theorem 2, gives the following remainder estimate for \(p\geq2\):

Here, using Theorem 3, we also obtain (?? ) with different remainder terms for \(1<p<2\leq n\):

\[\begin{align} &\nabla_{\gamma}\cdot\left(\left(\left(\frac{Q-p\theta}{p}\right)^{p}\frac{|\nabla_{\gamma}\rho|^{\alpha+p}}{\rho^{p\theta}}\right)^{\frac{p-1}{p}}\left(\frac{|\nabla_{\gamma}\rho|^{\alpha}}{\rho^{p(\theta-1)}}\right)^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right) \nonumber \\& = \nabla_{\gamma}\cdot\left(\left(\frac{Q-p\theta}{p}\right)^{p-1}\frac{|\nabla_{\gamma}\rho|^{\frac{(\alpha+p)(p-1)}{p}}}{\rho^{\theta(p-1)}}\frac{|\nabla_{\gamma}\rho|^{\frac{\alpha}{p}}}{\rho^{(\theta-1)}}\frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right) \nonumber \\& = \left(\frac{Q-p\theta}{p}\right)^{p-1}\nabla_{\gamma}\cdot\left(\frac{|\nabla_{\gamma}\rho|^{\alpha+p-2}}{\rho^{p\theta-1}}\nabla_{\gamma}\rho\right) \nonumber \\& = \left(\frac{Q-p\theta}{p}\right)^{p-1}\nabla_{\gamma}\cdot\left(\frac{|x|^{{\gamma}(\alpha+p-2)}}{\rho^{\gamma(\alpha+p-2)+p\theta-1}}\nabla_{\gamma}\rho\right) \nonumber \\& = \left(\frac{Q-p\theta}{p}\right)^{p-1}\nabla_{\gamma}\cdot\left(|x|^{\gamma(\alpha+p-2)}\rho^{-\gamma(\alpha+p-2)-p\theta+1}\nabla_{\gamma}\rho\right) .\label{darca32proof} \end{align}\tag{25}\] Applying formula (17 ) in (25 ) with \(s=\gamma(\alpha+p-2)\) and \(c=-\gamma(\alpha+p-2)-p\theta+1\), we obtain \[\begin{align} &\left(\frac{Q-p\theta}{p}\right)^{p-1}\nabla_{\gamma}\cdot\left(|x|^{\gamma(\alpha+p-2)}\rho^{-\gamma(\alpha+p-2)-p\theta+1}\nabla_{\gamma}\rho\right) = p\left(\frac{Q-p\theta}{p}\right)^{p}\frac{|x|^{\gamma(\alpha+p)}}{\rho^{\gamma(\alpha+p)+p\theta}} \\&=p\left(\frac{Q-p\theta}{p}\right)^{p}\frac{|\nabla_{\gamma}\rho|^{\alpha+p}}{\rho^{p\theta}}=pw + 0. \end{align}\] The condition (?? ) is satisfied with \(\phi=0\), which implies the desired results (?? ) and (?? ). The function \(h=\rho^{\frac{Q-p\theta}{p}}\) satisfies the Hölder equality condition, \[\begin{align} \left(\frac{p}{Q-p\theta}\right)^{p}\frac{|\nabla_\gamma \rho|^{\alpha}}{\rho^{p(\theta - 1)}}\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}h|^p}{|\nabla_{\gamma}\rho|^p} = \frac{\left| \nabla_\gamma \rho \right|^{\alpha + p}}{\rho^{p\theta}}|h|^{p}, \end{align}\] implying the sharpness in the inequality (?? ). \(\square\)

Now consider the following pair:

\[\begin{align} v=\left(\log \frac{R}{\rho}\right)^{\alpha+p},\quad w=\left(\frac{|\alpha+1|}{p}\right)^{p}\left(\log \frac{R}{\rho}\right)^{\alpha}\frac{|x|^{\gamma p}}{\rho^{\gamma p+p}}. \end{align}\] Then, we are able to obtain the following refinements of the power logarithmic \(L^p\)-Hardy type inequality D39A05?:

\[\begin{align} &\nabla_{\gamma}\cdot \left(\left(\left(\frac{|\alpha+1|}{p}\right)^{p}\left(\log \frac{R}{\rho}\right)^{\alpha}\frac{|x|^{\gamma p}}{\rho^{\gamma p+p}}\right)^{\frac{p-1}{p}}\left(\left(\log \frac{R}{\rho}\right)^{\alpha+p}\right)^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right) \nonumber \\&=\nabla_{\gamma}\cdot\left(\Phi \frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right) \nonumber \\&=\nabla_{\gamma}\Phi \cdot \frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}+\Phi \nabla_{\gamma}\cdot \left(\frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right) \nonumber \\&=\frac{1}{|\nabla_{\gamma}\rho|}\nabla_{\gamma}\Phi \cdot \nabla_{\gamma}\rho + \Phi\nabla_{\gamma}\cdot \left(\frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right) \nonumber \\&=T_{1}+T_{2}, \label{cor432t143t2} \end{align}\tag{26}\] where \[\begin{align} &T_1=\frac{1}{|\nabla_{\gamma}\rho|}\nabla_{\gamma}\Phi \cdot \nabla_{\gamma}\rho, \nonumber \\&T_2=\Phi\nabla_{\gamma}\cdot \left(\frac{\nabla_{\gamma}\rho}{|\nabla_{\gamma}\rho|}\right) \label{cor432t2} \end{align}\tag{27}\] with \[\begin{align} \Phi=\left(\frac{|\alpha+1|}{p}\right)^{p-1}\left(\log \frac{R}{\rho}\right)^{\alpha+1}\frac{|x|^{\gamma(p-1)}}{\rho^{(\gamma+1)(p-1)}}. \end{align}\] By using formula (18 ) in (27 ), we get \[\begin{align} &T_{2}=\Phi\cdot\left[(Q-1)\frac{|x|^{\gamma}}{\rho^{\gamma+1}}\right] \nonumber \\&=\left(\frac{|\alpha+1|}{p}\right)^{p-1}\left(\log \frac{R}{\rho}\right)^{\alpha+1}\frac{|x|^{\gamma(p-1)}}{\rho^{(\gamma+1)(p-1)}}\cdot(Q-1)\frac{|x|^{\gamma}}{\rho^{\gamma+1}} \nonumber \\&=\left(\frac{|\alpha+1|}{p}\right)^{p-1}(Q-1)\left(\log \frac{R}{\rho}\right)^{\alpha+1}\frac{|x|^{\gamma p}}{\rho^{(\gamma+1)p}}. \label{cor432t2951} \end{align}\tag{28}\] Now we compute \(T_1\): \[\begin{align} &T_{1}=\frac{1}{|\nabla_{\gamma}\rho|}(\nabla_{\gamma}\Phi)\cdot\nabla_{\gamma}\rho \nonumber \\&=\frac{1}{|\nabla_{\gamma}\rho|}\left(\frac{|\alpha+1|}{p}\right)^{p}\nabla_{\gamma}\left(\left(\log\frac{R}{\rho}\right)^{\alpha+1}|x|^{\gamma(p-1)}\rho^{-(\gamma+1)(p-1)}\right)\cdot\nabla_{\gamma}\rho \nonumber \\&=-\frac{\rho^{\gamma}}{|x|^{\gamma}}\left(\frac{|\alpha+1|}{p}\right)^{p}\frac{|x|^{\gamma(p+1)}}{\rho^{p(\gamma+1)+\gamma}}\left(\log \frac{R}{\rho}\right)^{\alpha}\left[(\alpha+1)+(p-1)\log \frac{R}{\rho}\right] \nonumber \\&=-\left(\frac{|\alpha+1|}{p}\right)^{p-1}\frac{|x|^{\gamma p}}{\rho^{p(\gamma+1)}}\left(\log \frac{R}{\rho}\right)^{\alpha}\left[(\alpha+1)+(p-1)\log \frac{R}{\rho}\right]. \label{cor432t1951} \end{align}\tag{29}\] Putting (28 ) and (29 ) to (26 ), one obtains \[\begin{align} &T_{1}+T_{2}=-\left(\frac{|\alpha+1|}{p}\right)^{p-1}\frac{|x|^{\gamma p}}{\rho^{p(\gamma+1)}}\left(\log \frac{R}{\rho}\right)^{\alpha}\left[(\alpha+1)+(p-1)\log \frac{R}{\rho}\right] \nonumber \\&+\left(\frac{|\alpha+1|}{p}\right)^{p-1}(Q-1)\left(\log \frac{R}{\rho}\right)^{\alpha+1}\frac{|x|^{\gamma p}}{\rho^{(\gamma+1)p}} \nonumber \\&=\left(\frac{|\alpha+1|}{p}\right)^{p-1}\frac{|x|^{\gamma p}}{\rho^{(\gamma+1)p}}\left(\log \frac{R}{\rho}\right)^{\alpha}\left[(Q-1)\log \frac{R}{\rho}-(p-1)\log \frac{R}{\rho}-(\alpha+1)\right] \nonumber \\&=\left(\frac{|\alpha+1|}{p}\right)^{p-1}\frac{|x|^{\gamma p}}{\rho^{(\gamma+1)p}}\left(\log \frac{R}{\rho}\right)^{\alpha}\left[(Q-p)\log \frac{R}{\rho}-(\alpha+1)\right] \nonumber \\&=\left(\frac{|\alpha+1|}{p}\right)^{p-1}\frac{|x|^{\gamma p}}{\rho^{(\gamma+1)p}}\left(\log \frac{R}{\rho}\right)^{\alpha}\left[(Q-p)\log \frac{R}{\rho}+|\alpha+1|\right] \nonumber \\&=p\left(\frac{|\alpha+1|}{p}\right)^{p}\frac{|x|^{\gamma p}}{\rho^{\gamma p+p}}\left(\log \frac{R}{\rho}\right)^{\alpha} \nonumber \\&+\left(\frac{|\alpha+1|}{p}\right)^{p-1}(Q-p)\frac{|x|^{\gamma p}}{\rho^{\gamma p+p}}\left(\log \frac{R}{\rho}\right)^{\alpha+1} \nonumber \\&=pw+\left(\frac{|\alpha+1|}{p}\right)^{p-1}(Q-p)\frac{|x|^{\gamma p}}{\rho^{\gamma p+p}}\left(\log \frac{R}{\rho}\right)^{\alpha+1}. \end{align}\] Therefore, \(\phi=\left(\frac{|\alpha+1|}{p}\right)^{p-1}(Q-p)\frac{|x|^{\gamma p}}{\rho^{\gamma p+p}}\left(\log \frac{R}{\rho}\right)^{\alpha+1}\geq0\). The condition (?? ) is satisfied and the results (?? ) and (?? ) are obtained. Since the function \(h=\left(\log \frac{R}{\rho}\right)^{-\frac{|\alpha+1|}{p}}\) satisfies the Hölder equality condition, \[\begin{align} \left(\frac{p}{|\alpha+1|}\right)^{p}\left(\log\frac{R}{\rho}\right)^{\alpha+p}\frac{|\nabla_{\gamma}\rho\cdot\nabla_{\gamma}h|^p}{|\nabla_{\gamma}\rho|^p} = \left(\log \frac{R}{\rho}\right)^{\alpha}\frac{|x|^{\gamma p}}{\rho^{\gamma p+p}}|h|^{p}, \end{align}\] the constant in the inequality (?? ) is sharp. \(\square\)

4.2 Caffarelli-Kohn-Nirenberg and Heisenberg-Pauli-Weyl type inequalities↩︎

\[\begin{align} \left\lVert w^{b}f\right\rVert_{L^{q}(\Omega)}\leq\left\lVert w^{{b}}f\right\rVert_{L^{q}(\Omega)}. \end{align}\] Case \(\delta=1\). In this case, we get \(p=r\) and \(c=\frac{1}{p}\) with \[\begin{gather} \left\lVert w^{\frac{1}{p}}f\right\rVert^{p}_{L^{p}(\Omega)}\leq\left\lVert v^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f}{|\nabla_{\gamma}\rho|}\right\rVert^{p}_{L^{p}(\Omega)}\\-\int_{\Omega}C_{p}\left(v^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|},v^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|} + w^{\frac{1}{p}}f\right)dz, \end{gather}\] which is (?? ) by Theorem 1.

\[\begin{align} \left\lVert w^{c}f\right\rVert_{L^{r}(\Omega)}=\left(\int_{\Omega}w^{cr}|f|^rdz\right)^{\frac{1}{r}}=\left(\int_{\Omega}\frac{|f|^{\delta r}}{w^{-\frac{\delta r}{p}}}\cdot\frac{|f|^{(1-\delta) r}}{w^{-b(1-\delta)r}}dz\right)^{\frac{1}{r}}. \end{align}\] Since \(\delta\in (0,1)\cap\left[\frac{r-q}{q},\frac{p}{r}\right]\) and \(p+q\geq r\), then using the Hölder inequality for \(\frac{\delta r}{p} + \frac{(1-\delta) r}{q} = 1\), we derive \[\begin{align} \left\lVert w^{c}f\right\rVert_{L^{r}(\Omega)}&\leq\left(\int_{\Omega}w|f|^{p}\right)^\frac{\delta}{p}\left(\int_{\Omega}w^{bq}|f|^{q}\right)^{\frac{1-\delta}{q}} \nonumber \\&=\left\lVert w^{\frac{1}{p}}f\right\rVert_{L^{p}(\Omega)}^{\delta}\left\lVert w^{b}f\right\rVert^{1-\delta}_{L^{q}(\Omega)} \nonumber \\&\leq\Biggl[\left\lVert v^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho\cdot\nabla_{\gamma}f}{|\nabla_{\gamma}\rho|}\right\rVert^{p}_{L^{p}(\Omega)} \nonumber \\&-\int_{\Omega}C_{p}\left(v^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|},v^{\frac{1}{p}}\frac{\nabla_{\gamma}\rho \cdot \nabla_{\gamma}f}{|\nabla_{\gamma}\rho|} + w^{\frac{1}{p}}f\right)dz\Biggr]^{\frac{\delta}{p}}\left\lVert w^{b}f\right\rVert^{1-\delta}_{L^{q}(\Omega)}. \label{ckn32last32eq} \end{align}\tag{30}\] Rewriting (30 ), we derive (?? ). \(\square\)

By the same technique, we can apply Corollaries 8, 9, 15 and 19 to obtain the corresponding CKN type inequalities.

In the special case, we also obtain some HPW type inequalities.

Data availability Data sharing not applicable to this article as no datasets were generated or analyzed during the current study

Declarations

Conflict of interest The authors declared that they have no conflict of interest to this work.

References↩︎